Comparison of leg dynamic models for quadrupedal robots with compliant backbone

Many quadrupeds are capable of power efficient gaits, especially trot and gallop, thanks to their flexible trunk. The oscillations of the system that includes the backbone, the tendons and musculature, store and release elastic energy, helping a smooth deceleration and a fast acceleration of the hindquarters and forequarters, which improves the dynamics of running and its energy efficiency. Forelegs and hindlegs play a key role in generating the bending moment in the trunk. In this paper we present our studies aimed at modeling and reproducing such phenomena for efficient quadrupedal robot locomotion. We propose a model, called mass-mass-spring model, that overcomes the limitation of existing models, and demonstrate that it allows studying how the masses of the legs generate a flexing force that helps the natural bending of the trunk during gallop. We apply our model to six animals, that adopt two different galloping patterns (called transverse and rotatory), and compare their energy efficiency.

www.nature.com/scientificreports/ also in 19,20 and 21 for hopping robots with compliant legs, and on flexible backbones for fish-like robots 22 . Also, a similar effect is also present in insects and birds, whose thoraxes contain compliant structures that accumulate and release energy during the flapping cycle at the benefit of consumption, and also flight stability, see, e.g. 23,24 .
In our current work we are investigating flexible backbones for legged robots, and how this can be exploited for fast and energy efficient running for quadruped robots. We have shown how a flexible spine can greatly help to achieve low power consumption storing and releasing energy during gait. Additionally, a dramatic energy saving can be obtained when the oscillations of the trunk reach a quasi-resonant regime 25 .
It is important to highlight that in most rigid-bodied running quadrupedal robots the legs mass is considered negligible for the purpose of studying robots body dynamics. Such assumption allows a simpler dynamic modeling, and light legs allow faster movements, and therefore faster running. However, when it comes to flexible trunk, this plays a key role, since the motion of the masses of the legs (plus tail and head) and their contact with the ground generates the trunk bending 26,27 .
In a running gait, the center of mass of the leg reaches its lowest point at the middle of step. The kinetic energy and gravitational potential energy reaction force are stored as elastic energy during the stance phase, when the leg touches the ground, and recovered during the flight phase, when the leg leaves the ground (see Fig. 1, steps h, a, b). The most studied model for robotic legs for running gaits is the SLIP (Spring Loaded Inverted Pendulum) model (see Fig. 2, top). In the flight phase the spring has no effect and thus is not considered, so the dynamics of the leg is represented taking into account only the point mass.
One of the first works on the SLIP model is due to Marc Raibert 28 , who showed that SLIP can describe the characteristics of running, trotting or hopping in one leg for bipeds and quadrupeds. Aspects such as stability,   www.nature.com/scientificreports/ dynamics and energy efficiency can be taken into account in this model. Also, Fumiya Iida et al. 29 showed that walking can be described using the bipedal version of this model. Using the SLIP model, various types of quadruped robots have been development, such as KOLT 7 , Scout II 8 , BigDog 9 , Start1ETH 10 and the MIT cheetah 30 , which are capable of walking, trotting and galloping at high speeds in different terrains. However, the SLIP model characterizes the dynamic formulation in a simple way, since it represents the robot's leg as a point mass and a massless spring that extends towards to the ground. This neglects the inertia of the leg 31 . Hence, this model falls short when it comes to the flight phase of the legs of a galloping robot, i.e. when the leg is not in stance with the ground, since it does not allow generating a force for bending the trunk. In most animals the mass of the leg is very important when performing the galloping movement, especially in quadrupeds with flexible trunks 26 . As mentioned earlier, the mass of the legs helps bending the trunk, allowing it to store and release elastic energy, which allows smoother movements and a more energy efficient gallop 25,27,32 . Therefore, in order to study the effect of the legs' masses in the dynamics, new models need to be developed. In this paper, we propose a Mass-Mass-Spring (MMS) leg model, as an alternative for quadruped robots with flexible trunk, and demonstrate that considering the mass of the leg in its dynamic modeling, it is possible to control the rotational force at the hip, and therefore induce a bending moment at the end of the trunk in the flight phase.
The motion of the leg has two main phases. The stance phase represents the system dynamics while it is in contact with the ground and where the spring k, which joins the mass m of the leg with the ground, acts passively (Fig. 2). Then, the flight phase represents the system dynamic during the flight, with the spring resting as the mass m. These two phases alternate in time, achieving a continuous movement pattern that causes the system to move along the x and y coordinates (Fig. 3).
In the following we compare the proposed Mass-Mass-Spring (MMS) leg model with the Spring Loaded Inverted Pendulum (SLIP) model. As can be seen in Fig. 2 (top), the SLIP model is composed of a point mass, M, which represents the hip, and a linear spring, k, that transmits the reaction forces between the ground and the hip, acting as energy storage during the stance phase. The kinematic and dynamic analyses are explained in the Appendix (see also [33][34][35][36] ). In addition to the mass of the hip, the MMS model takes into account also the mass of the leg, located at the knee joint, and a spring that establishes the contact between the mass of the leg and the ground, as shown in Fig. 2 (bottom). Thanks to the additional mass m, it is possible to model the forces that allow the trunk to bend in the flight phase.
The Mass-mass-spring model. The dynamic equations that represent the model for stance phase in horizontal and vertical axes are: where ẍ m and ẍ M are the horizontal acceleration and ÿ m and ÿ M are the vertical acceleration of the knee mass, m, and hip mass, M, respectively, for the stance phase.
The dynamic equations that represent the model for flight phase in horizontal and vertical axes are:

Methods
In order to assess the impact that the mass of the leg has on the spine of a quadruped legged robot, we considered the two different types of gallop, the so-called "rotary" and "transverse" gallop 26 .
The rotary gallop is employed, e.g., by the cheetah 26,37 , while the transverse gallop is employed by the horse 38 . In both modes it can be observed how the mass of the leg affects the bending of the trunk. To compare these kinds of gallop we analyzed six species of animals, three for each type gallops: cheeta, greyhound and lynx for rotatory gallop, and horse, antelope and alpaca for transverse gallop 39 . Figure 1 shows the differences between the two gallops. The rotatory gallop starts with a footfall of one of the forelimbs, to later support with the contralateral forelimb. After that, the flight phase starts, where the legs are completely below itself, and then continue a footfall with the ipsilateral hindlimb and support with the contralateral hindlimb, and then carry out another flight phase where the legs are extended. In contrast, the transverse gallop starts with one footfall of the hindlimbs and later to support with the contralateral hindlimb, following by the contralateral support of the ipsilateral forelimb and later a footfall of the contralateral forelimb, and then carry out flight phase, where the legs are completely below the body 26,39 . Figure 4 (bottom-left), shows the diagram of the simulation setup for measuring the energy that can be stored in the trunk thanks to the mass of the leg, located in the knee joint. The leg is attached at one end of a beam, while the other end of the beam was fixed. The aim of this setup is to measure the forces that are exerted at the ends of the beam representing the trunk of the robot.
In order to compare the MMS and the SLIP model, the following Key Performance Indicators (KPIs) were used: stride length and hopping height, which are used to evaluate the performance of quadruped robots 30,31 , and normal effort, tangential stress, minimum bending moment, maximum bending moment, and bending moment difference, which are used to know the stresses in the beam 40,41 . Here, the most important indicators are the minimum bending moment, maximum bending moment (see Fig. 5, right), that induce the bending at the two endings of the spine.  www.nature.com/scientificreports/ For the purpose of comparison, the same physical dimensions have been adopted for the two motion patterns, (see Table 1). The physical properties of the hind leg of a cheetah Acinonyx Jubatuss, obtained from 42 , were taken into account. In order to assess the effectiveness of the two models, we carried out simulations of a galloping cheetah, greyhound and lynx (rotatory gallop) and horse, antelope and alpaca (transverse gallop), using the kinematics and dynamics equations of the models. To acquire the movements described by the hip, knee and foot we analyzed footage of the running sequence of the two animals ("Maverick Galopp", 2012, https:// youtu. be/ iWGKO HeSpE0; "The Science of a Cheetah's Speed"-National Geographic and the Cincinnati Zoo, 2013, https:// youtu. be/ icFMT B0Pi0g) using the Tracker software (Tracker is a video analysis and modeling tool built on the Open Source Physics (OSP),https:// physl ets. org/ track er/) for the stance and flight phases (Fig. 5). With the data acquired, the angles α, ß and γ are calculated. These will be the inputs for each of the simulation models. Figure 6 depicts the gamma angles for SLIP model, and Fig. 7 shows the α and ß for MMS model. To perform the simulation, and to visualize of the movements, the Robotics toolbox system of MathWorks was used.

Results
Tables 2, 3 and 4 report the results of the cheetah, greyhound and lynx simulations. Although the stride length is bigger in the MMS model and the hopping is slightly bigger in SLIP model, it is important to note the difference in the minimum and maximum bending moments of the two models. In particular, a minimum bending moment is obtained with the MMS mode, while this is null in the SLIP model. Hence, the MMS model is capable of generating a more accurate bending of the trunk (Fig. 8). This is because it is the minimum bending moment that causes a bending of the trunk, similar to one of the cheetah galloping. The SLIP model only allows to obtain a buckling in the spine.    Tables 5, 6 and 7. As for the cheetah, in the trajectory of the horse it can be noticed an increase in the minimum bending moment and a decrease in the maximum bending moment for the MMS model, compared to the SLIP model. Again, this is due to the mass of the leg, which induces a rotational force in the hip, which in turn, causes a bending moment to be generated in the spine.   www.nature.com/scientificreports/ It can be noted that for the "bending difference" parameter (the difference between minimum and maximum bending,) the results are similar in the two models, but the SLIP model produces no "minimum bending". This is because in the MMS model the mass of the leg is taken into account (about 6% of the total mass), generating the bending moment at the end of the spine. In MMS A lower maximum bending at the fixed end of spine, is generated. Note that the total mass is equal in the two models (M + m in MMS, M total in SLIP).
The tangential stress parameter is useful because it helps to know what force stretches the spine in galloping. This is similar for the two models, despite the masses are distributed between hip and knee in the MMS model while it is concentrated at the hip in the SLIP model. Figures 9 and 10 show the details of the displacement and velocity of the hip and knee resulting from the simulation of stance phase (solid line) and flight phase (dashed line) for the cheetah and the horse, considering the parameters of Table 1. We choose these two anymals as representative of their respective groups, because < < the transverse gallop is epitomized by the horse and the rotary gallop by the cheetah > > 26 . Figures 9a,c and 10a,c depict the movement of the hip, in MMS and SLIP models, describes a parabolic movement in the flight phase, since the heaviest mass is at the hip drives the trajectory of the leg in the flight phase. Similarly, as can be seen in Figs. 9b and 10b, there are abrupt changes in velocities, both in stance phase and flight phase. These are due to the mass in the knee which allows to obtain a moment of force at the end of the spine that helps to accumulate energy during flight phase, and releases energy during the stance phase.

Discussion
The purpose of our current work is to understand and model the mechanics of bending of the trunk in quadrupeds with flexible trunk, with the aim of reproducing the mechanism of storing/releasing energy during galloping. This is a key feature for improving the performance of quadruped robots. Additionally, a flexible trunk helps a  www.nature.com/scientificreports/ smooth deceleration and a fast acceleration of the different parts of the body involved during running, as well as it reduces shocks in the mechanical structure at the benefit of lighter structures and smoother movements. We have demonstrated how using the modeling the gallop of a quadruped robot with flexible trunk by using the proposed MMS leg model produces better results with respect to the commonly used SLIP leg model. This model allows a moment of force to be generated at the center of mass of the hip when performing the galloping movement, called minimum bending moment. This is because in the MMS model the mass of the leg is taken into account (about 6% of the total mass), generating the bending moment at the end of the spine. Note that the total mass is equal in the two models (M + m in MMS, M total in SLIP). Due to its mathematical formulation, the SLIP model cannot reproduce the bending moment at the free end of the spine. This is precisely the reason we developed the MMS model.
Even if the purpose of the paper is not a comparative analysis of biological data, but to assess the goodness of the proposed mathematical model, some conclusion from the experiments can be drawn, keeping in mind that the small number of animals analyzed does not allow an in-depth statistical analysis.
Comparing the KPIs of the six animals analyzed, it can be seen that in general, the animals that employ rotary gallop have a higher bending moment, while a transversal gallop produces a higher maximum bending moment ( Table 8). The transversal gait begins positioning the hind legs on the ground and the front legs in the flight phase, generating a rotational force that allows generating a maximum bending moment higher than the minimum bending moment on its hind limb. On the contrary, the rotary gait begins positioning front legs on the ground and the hind legs in the flight phase, generating a high rotational force in its rear part and consequently a bigger value of the minimum bending moment, allowing a bigger flexion in the trunk and therefore a